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#[global] Unset Universe Checking.

(*******************************************************************

 Limits in bicategories of structured categories

 We look at terminal objects, products, pullbacks, and
 Eilenberg-Moore objects

 Contents
 1. Limits of categories with a terminal objects
 2. Limits of categories with products
 3. Limits of categories with pullbacks
 4. Limits of categories with finite limits
 5. Limits of categories with initial objects
 6. Limits of categories with coproducts

 *******************************************************************)
Notation "_ + _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ - _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ * _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ = _" was already used in scope type_scope. [notation-overridden,parsing,default]
Notation "_ <-> _" was already used in scope type_scope. [notation-overridden,parsing,default]
Notation "_ > _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ < _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ <= _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ >= _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ ≠ _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ ≠ _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Categories.StandardCategories. Require Import UniMath.CategoryTheory.Categories.EilenbergMoore. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.IsoCommaCategory. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.Examples.UnitCategoryLimits. Require Import UniMath.CategoryTheory.Limits.Examples.CategoryProductLimits. Require Import UniMath.CategoryTheory.Limits.Examples.IsoCommaLimits. Require Import UniMath.CategoryTheory.Limits.Examples.EilenbergMooreLimits. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.Core.Examples.StructuredCategories. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Sub1Cell. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Prod. Require Import UniMath.Bicategories.DisplayedBicats.Examples.MonadsLax. Require Import UniMath.Bicategories.Limits.Final. Require Import UniMath.Bicategories.Limits.Products. Require Import UniMath.Bicategories.Limits.Pullbacks. Require Import UniMath.Bicategories.Limits.EilenbergMooreObjects. Require Import UniMath.Bicategories.Limits.Examples.BicatOfUnivCatsLimits. Require Import UniMath.Bicategories.Limits.Examples.TotalBicategoryLimits. Require Import UniMath.Bicategories.Limits.Examples.DispConstructionsLimits. Require Import UniMath.Bicategories.Limits.Examples.SubbicatLimits. Require Import UniMath.Bicategories.Monads.Examples.MonadsInTotalBicat. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Examples.MonadInclusion. Local Open Scope cat. (** 1. Limits of categories with a terminal objects *)

disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats

disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats

(λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 bifinal_cats)

∏ x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 x) (pr1 bifinal_cats) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))

(λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 bifinal_cats)
exact terminal_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 x) (pr1 bifinal_cats) (pr12 x) terminal_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
C: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 C) (pr1 bifinal_cats) (pr12 C) terminal_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 C))
apply functor_to_unit_preserves_terminal. Defined.

bifinal_obj univ_cat_with_terminal_obj

bifinal_obj univ_cat_with_terminal_obj

bifinal_obj bicat_of_univ_cats

(λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 ?HB)

∏ x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 x) (pr1 ?HB) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 ?HB) (pr1 x))

bifinal_obj bicat_of_univ_cats
exact bifinal_cats.

(λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 bifinal_cats)
exact terminal_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 x) (pr1 bifinal_cats) (pr12 x) terminal_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
C: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 C) (pr1 bifinal_cats) (pr12 C) terminal_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 C))
apply functor_to_unit_preserves_terminal. Defined.

disp_has_binprod disp_bicat_terminal_obj has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_terminal_obj has_binprod_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂, terminal_category_binproduct (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr1_preserves_terminal.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr2_preserves_terminal.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

preserves_terminal (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂
preserves_terminal (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

preserves_terminal (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

preserves_terminal (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

has_binprod univ_cat_with_terminal_obj

has_binprod univ_cat_with_terminal_obj

has_binprod bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (?HB (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂, terminal_category_binproduct (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr1_preserves_terminal.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr2_preserves_terminal.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)), terminal_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

preserves_terminal (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂
preserves_terminal (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

preserves_terminal (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)
q: binprod_cone C₁ C₂

preserves_terminal (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

disp_has_pb disp_bicat_terminal_obj has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_terminal_obj has_pb_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) q))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
exact (λ C₁ C₂ C₃ F G, terminal_category_iso_comma _ _ (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₂, C₃ ⟧), terminal_category_iso_comma (pr1 (pr1 F,, pr12 F)) (pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G, iso_comma_pr1_preserves_terminal _ _ (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₂, C₃ ⟧), terminal_category_iso_comma (pr1 (pr1 F,, pr12 F)) (pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G, iso_comma_pr2_preserves_terminal _ _ (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₂, C₃ ⟧), terminal_category_iso_comma (pr1 (pr1 F,, pr12 F)) (pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) q))
exact (λ C₁ C₂ C₃ F G q, iso_comma_ump1_preserves_terminal _ _ (pr22 G) _ (pr22 (pb_cone_pr1 q)) _ (pr22 (pb_cone_pr2 q)) _). Defined.

has_pb univ_cat_with_terminal_obj

has_pb univ_cat_with_terminal_obj

has_pb bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) q))

has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
exact (λ C₁ C₂ C₃ F G, terminal_category_iso_comma _ _ (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₂, C₃ ⟧), terminal_category_iso_comma (pr1 (pr1 F,, pr12 F)) (pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G, iso_comma_pr1_preserves_terminal _ _ (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₂, C₃ ⟧), terminal_category_iso_comma (pr1 (pr1 F,, pr12 F)) (pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
exact (λ C₁ C₂ C₃ F G, iso_comma_pr2_preserves_terminal _ _ (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z Py Pz g), composition_preserves_terminal HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g), composition_preserves_terminal HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_terminal F0) y0 z0 Py Pz g0), composition_preserves_terminal HF HG) ⟦ C₂, C₃ ⟧), terminal_category_iso_comma (pr1 (pr1 F,, pr12 F)) (pr1 (pr1 G,, pr12 G)) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y0 z0 Py Pz g0), composition_preserves_terminal HF HG)) q))
exact (λ C₁ C₂ C₃ F G q, iso_comma_ump1_preserves_terminal _ _ (pr22 G) _ (pr22 (pb_cone_pr1 q)) _ (pr22 (pb_cone_pr2 q)) _). Defined.

bicat_has_em univ_cat_with_terminal_obj

bicat_has_em univ_cat_with_terminal_obj

bicat_has_em bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr11 (?HB (pr1_of_mnd_total_bicat m)))

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr11 m) (?Hcone m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (?HB (pr1_of_mnd_total_bicat m)))))

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr121 q) (?Hcone m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (?HB (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) m q))

bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))
exact (λ m, terminal_eilenberg_moore_cat _ (pr12 (ob_of_mnd m))).

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr11 m) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), terminal_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0))) m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))))
exact (λ m, eilenberg_moore_pr_preserves_terminal _ (pr12 (ob_of_mnd m))).

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), terminal_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)))
q: em_cone m

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG))), terminal_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)) (pr12 (ob_of_mnd m))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)))
q: em_cone m

preserves_terminal (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Terminal (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Terminal (pr1 C0)) C), identity_preserves_terminal C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Terminal (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_terminal F) y z Py Pz g), composition_preserves_terminal HF HG)) m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))). Defined. (** 2. Limits of categories with products *)

disp_bifinal_obj disp_bicat_binprod bifinal_cats

disp_bifinal_obj disp_bicat_binprod bifinal_cats

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 bifinal_cats)

∏ x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 bifinal_cats)
exact binproduct_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) binproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
x: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) binproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
apply functor_to_unit_preserves_binproduct. Defined.

bifinal_obj univ_cat_with_binprod

bifinal_obj univ_cat_with_binprod

bifinal_obj bicat_of_univ_cats

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 ?HB)

∏ x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 x) (pr1 ?HB) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 ?HB) (pr1 x))

bifinal_obj bicat_of_univ_cats
exact bifinal_cats.

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 bifinal_cats)
exact binproduct_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) binproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
x: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) binproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
apply functor_to_unit_preserves_binproduct. Defined.

disp_has_binprod disp_bicat_binprod has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_binprod has_binprod_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

preserves_binproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂
preserves_binproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

preserves_binproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

preserves_binproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

has_binprod univ_cat_with_binprod

has_binprod univ_cat_with_binprod

has_binprod bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (?HB (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
BinProducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)), binproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

preserves_binproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂
preserves_binproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

preserves_binproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
q: binprod_cone C₁ C₂

preserves_binproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

disp_has_pb disp_bicat_binprod has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_binprod has_pb_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) q))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_binproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_binproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

has_pb univ_cat_with_binprod

has_pb univ_cat_with_binprod

has_pb bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) q))

has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
BinProducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧

BinProducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z Py Pz g), composition_preserves_binproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g), composition_preserves_binproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y0 z0 Py Pz g0), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_binproduct F0) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧), binproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_binproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_binproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_binproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinProducts (pr1 C₁)) (_ : BinProducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : BinProducts (pr1 C)), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinProducts (pr1 x)) (_ : BinProducts (pr1 y)) (_ : BinProducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_binproduct f) (HG : preserves_binproduct g), composition_preserves_binproduct HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

bicat_has_em univ_cat_with_binprod

bicat_has_em univ_cat_with_binprod

bicat_has_em bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr11 (?HB (pr1_of_mnd_total_bicat m)))

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr11 m) (?Hcone m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (?HB (pr1_of_mnd_total_bicat m)))))

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr121 q) (?Hcone m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (?HB (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) m q))

bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))
exact (λ m, BinProducts_eilenberg_moore_cat _ (pr12 (ob_of_mnd m))).

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr11 m) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), BinProducts_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0))) m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))))
exact (λ m, eilenberg_moore_pr_preserves_binproduct _ (pr12 (ob_of_mnd m))).

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), BinProducts_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)))
q: em_cone m

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG))), BinProducts_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)) (pr12 (ob_of_mnd m))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)))
q: em_cone m

preserves_binproduct (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinProducts (pr1 C0)) C), identity_preserves_binproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinProducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_binproduct F) y z Py Pz g), composition_preserves_binproduct HF HG)) m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))). Defined. (** 3. Limits of categories with pullbacks *)

disp_bifinal_obj disp_bicat_pullback bifinal_cats

disp_bifinal_obj disp_bicat_pullback bifinal_cats

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 bifinal_cats)

∏ x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 x) (pr1 bifinal_cats) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 bifinal_cats)
exact pullbacks_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 x) (pr1 bifinal_cats) (pr12 x) pullbacks_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
x: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 x) (pr1 bifinal_cats) (pr12 x) pullbacks_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
apply functor_to_unit_preserves_pullback. Defined.

bifinal_obj univ_cat_with_pb

bifinal_obj univ_cat_with_pb

bifinal_obj bicat_of_univ_cats

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 ?HB)

∏ x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 x) (pr1 ?HB) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 ?HB) (pr1 x))

bifinal_obj bicat_of_univ_cats
exact bifinal_cats.

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 bifinal_cats)
exact pullbacks_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 x) (pr1 bifinal_cats) (pr12 x) pullbacks_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
x: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 x) (pr1 bifinal_cats) (pr12 x) pullbacks_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
apply functor_to_unit_preserves_pullback. Defined.

disp_has_binprod disp_bicat_pullback has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_pullback has_binprod_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

preserves_pullback (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂
preserves_pullback (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

preserves_pullback (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

preserves_pullback (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

has_binprod univ_cat_with_pb

has_binprod univ_cat_with_pb

has_binprod bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (?HB (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
Pullbacks (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)), pullbacks_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

preserves_pullback (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂
preserves_pullback (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

preserves_pullback (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
q: binprod_cone C₁ C₂

preserves_pullback (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

disp_has_pb disp_bicat_pullback has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_pullback has_pb_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) q))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_pullback (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_pullback (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

has_pb univ_cat_with_pb

has_pb univ_cat_with_pb

has_pb bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) q))

has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
Pullbacks (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧

Pullbacks (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z Py Pz g), composition_preserves_pullback HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g), composition_preserves_pullback HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y0 z0 Py Pz g0), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y0 z0 Py Pz g0), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_pullback F0) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧), pullbacks_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_pullback (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_pullback (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_pullback (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Pullbacks (pr1 C₁)) (_ : Pullbacks (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : Pullbacks (pr1 C)), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (_ : Pullbacks (pr1 x)) (_ : Pullbacks (pr1 y)) (_ : Pullbacks (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_pullback f) (HG : preserves_pullback g), composition_preserves_pullback HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

bicat_has_em univ_cat_with_pb

bicat_has_em univ_cat_with_pb

bicat_has_em bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr11 (?HB (pr1_of_mnd_total_bicat m)))

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr11 m) (?Hcone m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (?HB (pr1_of_mnd_total_bicat m)))))

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr121 q) (?Hcone m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (?HB (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) m q))

bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))
exact (λ m, Pullbacks_eilenberg_moore _ (pr12 (ob_of_mnd m))).

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr11 m) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), Pullbacks_eilenberg_moore (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0))) m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))))
exact (λ m, eilenberg_moore_pr_preserves_pullback _ (pr12 (ob_of_mnd m))).

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), Pullbacks_eilenberg_moore (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)))
q: em_cone m

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG))), Pullbacks_eilenberg_moore (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)) (pr12 (ob_of_mnd m))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)))
q: em_cone m

preserves_pullback (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Pullbacks (pr1 C0)) C), identity_preserves_pullback C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Pullbacks (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_pullback F) y z Py Pz g), composition_preserves_pullback HF HG)) m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))). Defined. (** 4. Limits of categories with finite limits *)

disp_bifinal_obj disp_bicat_finlim bifinal_cats

disp_bifinal_obj disp_bicat_finlim bifinal_cats

disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats

disp_bifinal_obj disp_bicat_pullback bifinal_cats

disp_bifinal_obj disp_bicat_terminal_obj bifinal_cats
exact disp_bifinal_univ_cat_with_terminal_obj.

disp_bifinal_obj disp_bicat_pullback bifinal_cats
exact disp_bifinal_obj_univ_cat_with_pb. Defined.

bifinal_obj bicat_of_univ_cat_with_finlim

bifinal_obj bicat_of_univ_cat_with_finlim

disp_2cells_isaprop disp_bicat_finlim

∏ (x y : bicat_of_univ_cats) (f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g) (xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y) (ff : xx -->[ f] yy) (gg : xx -->[ g] yy), disp_2cells α ff gg

bifinal_obj bicat_of_univ_cats

disp_bifinal_obj disp_bicat_finlim ?HB

disp_2cells_isaprop disp_bicat_finlim

disp_2cells_isaprop disp_bicat_terminal_obj

disp_2cells_isaprop disp_bicat_pullback

disp_2cells_isaprop disp_bicat_terminal_obj
apply disp_2cells_isaprop_subbicat.

disp_2cells_isaprop disp_bicat_pullback
apply disp_2cells_isaprop_subbicat.

∏ (x y : bicat_of_univ_cats) (f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g) (xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y) (ff : xx -->[ f] yy) (gg : xx -->[ g] yy), disp_2cells α ff gg
x, y: bicat_of_univ_cats
f, g: bicat_of_univ_cats ⟦ x, y ⟧
α: f ==> g
xx: disp_bicat_finlim x
yy: disp_bicat_finlim y
ff: xx -->[ f] yy
gg: xx -->[ g] yy

disp_2cells α ff gg
exact ((tt ,, tt) ,, (tt ,, tt)).

bifinal_obj bicat_of_univ_cats
exact bifinal_cats.

disp_bifinal_obj disp_bicat_finlim bifinal_cats
exact disp_bifinal_obj_univ_cat_with_finlim. Defined.

disp_has_binprod disp_bicat_finlim has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_finlim has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_terminal_obj has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_pullback has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_terminal_obj has_binprod_bicat_of_univ_cats
exact disp_has_binprod_univ_cat_with_terminal_obj.

disp_has_binprod disp_bicat_pullback has_binprod_bicat_of_univ_cats
exact disp_has_binprod_univ_cat_with_pb. Defined.

has_binprod bicat_of_univ_cat_with_finlim

has_binprod bicat_of_univ_cat_with_finlim

disp_2cells_isaprop disp_bicat_finlim

∏ (x y : bicat_of_univ_cats) (f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g) (xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y) (ff : xx -->[ f] yy) (gg : xx -->[ g] yy), disp_2cells α ff gg

disp_locally_groupoid disp_bicat_finlim

has_binprod bicat_of_univ_cats

disp_has_binprod disp_bicat_finlim ?HB

disp_2cells_isaprop disp_bicat_finlim

disp_2cells_isaprop disp_bicat_terminal_obj

disp_2cells_isaprop disp_bicat_pullback

disp_2cells_isaprop disp_bicat_terminal_obj
apply disp_2cells_isaprop_subbicat.

disp_2cells_isaprop disp_bicat_pullback
apply disp_2cells_isaprop_subbicat.

∏ (x y : bicat_of_univ_cats) (f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g) (xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y) (ff : xx -->[ f] yy) (gg : xx -->[ g] yy), disp_2cells α ff gg
x, y: bicat_of_univ_cats
f, g: bicat_of_univ_cats ⟦ x, y ⟧
α: f ==> g
xx: disp_bicat_finlim x
yy: disp_bicat_finlim y
ff: xx -->[ f] yy
gg: xx -->[ g] yy

disp_2cells α ff gg
exact ((tt ,, tt) ,, (tt ,, tt)).

disp_locally_groupoid disp_bicat_finlim

disp_locally_groupoid disp_bicat_terminal_obj

disp_locally_groupoid disp_bicat_pullback

disp_locally_groupoid disp_bicat_terminal_obj

is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.

disp_locally_groupoid disp_bicat_pullback

is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.

has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.

disp_has_binprod disp_bicat_finlim has_binprod_bicat_of_univ_cats
exact disp_has_binprod_univ_cat_with_finlim. Defined.

disp_has_pb disp_bicat_finlim has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_finlim has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_terminal_obj has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_pullback has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_terminal_obj has_pb_bicat_of_univ_cats
exact disp_has_pb_univ_cat_with_terminal_obj.

disp_has_pb disp_bicat_pullback has_pb_bicat_of_univ_cats
exact disp_has_pb_univ_cat_with_pb. Defined.

has_pb bicat_of_univ_cat_with_finlim

has_pb bicat_of_univ_cat_with_finlim

disp_2cells_isaprop disp_bicat_finlim

∏ (x y : bicat_of_univ_cats) (f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g) (xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y) (ff : xx -->[ f] yy) (gg : xx -->[ g] yy), disp_2cells α ff gg

disp_locally_groupoid disp_bicat_finlim

has_pb bicat_of_univ_cats

disp_has_pb disp_bicat_finlim ?HB

disp_2cells_isaprop disp_bicat_finlim

disp_2cells_isaprop disp_bicat_terminal_obj

disp_2cells_isaprop disp_bicat_pullback

disp_2cells_isaprop disp_bicat_terminal_obj
apply disp_2cells_isaprop_subbicat.

disp_2cells_isaprop disp_bicat_pullback
apply disp_2cells_isaprop_subbicat.

∏ (x y : bicat_of_univ_cats) (f g : bicat_of_univ_cats ⟦ x, y ⟧) (α : f ==> g) (xx : disp_bicat_finlim x) (yy : disp_bicat_finlim y) (ff : xx -->[ f] yy) (gg : xx -->[ g] yy), disp_2cells α ff gg
x, y: bicat_of_univ_cats
f, g: bicat_of_univ_cats ⟦ x, y ⟧
α: f ==> g
xx: disp_bicat_finlim x
yy: disp_bicat_finlim y
ff: xx -->[ f] yy
gg: xx -->[ g] yy

disp_2cells α ff gg
exact ((tt ,, tt) ,, (tt ,, tt)).

disp_locally_groupoid disp_bicat_finlim

disp_locally_groupoid disp_bicat_terminal_obj

disp_locally_groupoid disp_bicat_pullback

disp_locally_groupoid disp_bicat_terminal_obj

is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.

disp_locally_groupoid disp_bicat_pullback

is_univalent_2 bicat_of_univ_cats
apply univalent_cat_is_univalent_2.

has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.

disp_has_pb disp_bicat_finlim has_pb_bicat_of_univ_cats
exact disp_has_pb_univ_cat_with_finlim. Defined. (** 5. Limits of categories with initial objects *)

disp_bifinal_obj disp_bicat_initial_obj bifinal_cats

disp_bifinal_obj disp_bicat_initial_obj bifinal_cats

(λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 bifinal_cats)

∏ x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 x) (pr1 bifinal_cats) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))

(λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 bifinal_cats)
exact initial_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 x) (pr1 bifinal_cats) (pr12 x) initial_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
C: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 C) (pr1 bifinal_cats) (pr12 C) initial_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 C))
apply functor_to_unit_preserves_initial. Defined.

bifinal_obj univ_cat_with_initial

bifinal_obj univ_cat_with_initial

bifinal_obj bicat_of_univ_cats

(λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 ?HB)

∏ x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 x) (pr1 ?HB) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 ?HB) (pr1 x))

bifinal_obj bicat_of_univ_cats
exact bifinal_cats.

(λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 bifinal_cats)
exact initial_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 x) (pr1 bifinal_cats) (pr12 x) initial_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
C: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 C) (pr1 bifinal_cats) (pr12 C) initial_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 C))
apply functor_to_unit_preserves_initial. Defined.

disp_has_binprod disp_bicat_initial_obj has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_initial_obj has_binprod_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂, initial_category_binproduct (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr1_preserves_initial.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr2_preserves_initial.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

preserves_initial (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂
preserves_initial (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

preserves_initial (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

preserves_initial (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

has_binprod univ_cat_with_initial

has_binprod univ_cat_with_initial

has_binprod bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (?HB (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
exact (λ C₁ C₂, initial_category_binproduct (pr12 C₁) (pr12 C₂)).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr1_preserves_initial.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
apply pr2_preserves_initial.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)), initial_category_binproduct (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

preserves_initial (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂
preserves_initial (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

preserves_initial (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
q: binprod_cone C₁ C₂

preserves_initial (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

disp_has_pb disp_bicat_initial_obj has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_initial_obj has_pb_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) q))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_initial (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_initial (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

has_pb univ_cat_with_initial

has_pb univ_cat_with_initial

has_pb bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) q))

has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
preserves_initial (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
Initial (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

preserves_initial (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧

Initial (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z Py Pz g), composition_preserves_initial HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g), composition_preserves_initial HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y0 z0 Py Pz g0), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y0 z0 Py Pz g0), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_initial F0) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧), initial_category_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_initial (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_initial (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)
F: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_initial (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : Initial (pr1 C₁)) (_ : Initial (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : Initial (pr1 C)), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (_ : Initial (pr1 x)) (_ : Initial (pr1 y)) (_ : Initial (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_initial f) (HG : preserves_initial g), composition_preserves_initial HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

bicat_has_em univ_cat_with_initial

bicat_has_em univ_cat_with_initial

bicat_has_em bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr11 (?HB (pr1_of_mnd_total_bicat m)))

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr11 m) (?Hcone m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (?HB (pr1_of_mnd_total_bicat m)))))

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr121 q) (?Hcone m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (?HB (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) m q))

bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), (λ C : bicat_of_univ_cats, Initial (pr1 C)) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))

preserves_initial (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr11 m) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), initial_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))

preserves_initial (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), initial_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))
q: em_cone m

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG))), initial_eilenberg_moore_cat (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)) (pr12 (ob_of_mnd m)) (pr22 (endo_of_mnd m))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))
q: em_cone m

preserves_initial (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))
q: em_cone m
preserves_initial (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) m q)))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))
q: em_cone m

preserves_initial (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)))
q: em_cone m

preserves_initial (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, Initial (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, Initial (pr1 C0)) C), identity_preserves_initial C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, Initial (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, Initial (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, Initial (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, Initial (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_initial F) y z Py Pz g), composition_preserves_initial HF HG)) m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))). Defined. (** 6. Limits of categories with coproducts *)

disp_bifinal_obj disp_bicat_bincoprod bifinal_cats

disp_bifinal_obj disp_bicat_bincoprod bifinal_cats

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 bifinal_cats)

∏ x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 bifinal_cats)
exact bincoproduct_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) bincoproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
C: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 C) (pr1 bifinal_cats) (pr12 C) bincoproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 C))
apply functor_to_unit_preserves_bincoproduct. Defined.

bifinal_obj univ_cat_with_bincoprod

bifinal_obj univ_cat_with_bincoprod

bifinal_obj bicat_of_univ_cats

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 ?HB)

∏ x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 x) (pr1 ?HB) (pr12 x) ?P_final (is_bifinal_1cell_property (pr2 ?HB) (pr1 x))

bifinal_obj bicat_of_univ_cats
exact bifinal_cats.

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 bifinal_cats)
exact bincoproduct_unit_category.

∏ x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 x) (pr1 bifinal_cats) (pr12 x) bincoproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 x))
C: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 C) (pr1 bifinal_cats) (pr12 C) bincoproduct_unit_category (is_bifinal_1cell_property (pr2 bifinal_cats) (pr1 C))
apply functor_to_unit_preserves_bincoproduct. Defined.

disp_has_binprod disp_bicat_bincoprod has_binprod_bicat_of_univ_cats

disp_has_binprod disp_bicat_bincoprod has_binprod_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

preserves_bincoproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂
preserves_bincoproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

preserves_bincoproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

preserves_bincoproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

has_binprod univ_cat_with_bincoprod

has_binprod univ_cat_with_bincoprod

has_binprod bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 x) (?Hcone x y) (pr12 x) (binprod_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (?HB (pr1 x) (pr1 y))) (pr1 y) (?Hcone x y) (pr12 y) (binprod_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y))) (pr121 q) (?Hcone x y) (binprod_ump_1cell (pr2 (?HB (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))

has_binprod bicat_of_univ_cats
exact has_binprod_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 x) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₁) (binprod_cone_pr1 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (pr12 y) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (pr12 C₂) (binprod_cone_pr2 (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
BinCoproducts (pr1 C₂)
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (q : binprod_cone x y), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) x y) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 x) (pr1 y))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)), bincoproducts_in_product_category (pr12 C₁) (pr12 C₂)) C₁ C₂) (binprod_ump_1cell (pr2 (has_binprod_bicat_of_univ_cats (pr1 C₁) (pr1 C₂))) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q)))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

preserves_bincoproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂
preserves_bincoproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

preserves_bincoproduct (binprod_cone_pr1 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr1 q)).
C₁, C₂: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
q: binprod_cone C₁ C₂

preserves_bincoproduct (binprod_cone_pr2 (make_binprod_cone (pr1 q) (pr1 (binprod_cone_pr1 q)) (pr1 (binprod_cone_pr2 q))))
exact (pr22 (binprod_cone_pr2 q)). Defined.

disp_has_pb disp_bicat_bincoprod has_pb_bicat_of_univ_cats

disp_has_pb disp_bicat_bincoprod has_pb_bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) q))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_bincoproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_bincoproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_bincoproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_bincoproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_bincoproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

has_pb univ_cat_with_bincoprod

has_pb univ_cat_with_bincoprod

has_pb bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) (?Hcone x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) (?Hcone x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) (?Hcone x y z f g) (pb_ump_mor (pr2 (?HB (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) q))

has_pb bicat_of_univ_cats
exact has_pb_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

(λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G)))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_bincoproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 x) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 x) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₁) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₁) (pb_cone_pr1 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_bincoproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr1 y) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pr12 y) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr1 C₂) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pr12 C₂) (pb_cone_pr2 (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
preserves_bincoproduct (pr1 G)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₁)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
BinCoproducts (pr1 C₂)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

preserves_bincoproduct (pr1 G)
exact (pr22 G).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₁)
exact (pr12 C₁).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧

BinCoproducts (pr1 C₂)
exact (pr12 C₂).

∏ (x : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (y : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (z : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z Py Pz g), composition_preserves_bincoproduct HF HG)) (f : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ x, z ⟧) (g : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ y, z ⟧) (q : pb_cone f g), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x0 y0 Px Py f0) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) x y z f g) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 x) (pr1 y) (pr1 z) (pr1 f) (pr1 g))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x0 y0 z0 : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x0) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y0) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z0) (f0 : bicat_of_univ_cats ⟦ x0, y0 ⟧) (g0 : bicat_of_univ_cats ⟦ y0, z0 ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x0 y0 Px Py f0) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y0 z0 Py Pz g0), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr1 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (pr121 q) ((λ (C₁ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₂ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁0, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (C₃ : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) (F : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧) (G : subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) x y Px Py f) (HG : (λ (C₁0 C₂0 : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁0) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂0) (F0 : bicat_of_univ_cats ⟦ C₁0, C₂0 ⟧), preserves_bincoproduct F0) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧), bincoproducts_in_iso_comma (pr1 F) (pr1 G) (pr22 F) (pr22 G) (pr12 C₁) (pr12 C₂)) C₁ C₂ C₃ F G) (pb_ump_mor (pr2 (has_pb_bicat_of_univ_cats (pr1 C₁) (pr1 C₂) (pr1 C₃) (pr1 F) (pr1 G))) (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pr1 F)
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_bincoproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G
preserves_bincoproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pr1 F)
exact (pr22 F).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pb_cone_pr1 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
exact (pr22 (pb_cone_pr1 q)).
C₁, C₂, C₃: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)
F: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₁, C₃ ⟧
G: subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG) ⟦ C₂, C₃ ⟧
q: pb_cone F G

preserves_bincoproduct (pb_cone_pr2 (total_pb_cone_help_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C₁)) (_ : BinCoproducts (pr1 C₂)) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : BinCoproducts (pr1 C)), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (_ : BinCoproducts (pr1 x)) (_ : BinCoproducts (pr1 y)) (_ : BinCoproducts (pr1 z)) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : preserves_bincoproduct f) (HG : preserves_bincoproduct g), composition_preserves_bincoproduct HF HG)) q))
exact (pr22 (pb_cone_pr2 q)). Defined.

bicat_has_em univ_cat_with_bincoprod

bicat_has_em univ_cat_with_bincoprod

bicat_has_em bicat_of_univ_cats

is_univalent_2 bicat_of_univ_cats

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr11 (?HB (pr1_of_mnd_total_bicat m)))

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr11 m) (?Hcone m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (?HB (pr1_of_mnd_total_bicat m)))))

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr11 (?HB (pr1_of_mnd_total_bicat m))) (pr121 q) (?Hcone m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (?HB (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) m q))

bicat_has_em bicat_of_univ_cats
exact has_em_bicat_of_univ_cats.

is_univalent_2 bicat_of_univ_cats
exact univalent_cat_is_univalent_2.

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))

preserves_bincoproduct (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).

∏ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr11 m) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), bincoproducts_eilenberg_moore (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m) (pr121 m) (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m)))))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))

preserves_bincoproduct (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).

∏ (m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))) (q : em_cone m), (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m0 : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), bincoproducts_eilenberg_moore (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m0)) (pr12 (ob_of_mnd m0)) (pr22 (endo_of_mnd m0))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))
q: em_cone m

(λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (pr11 q) (pr11 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr121 q) ((λ m : mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG))), bincoproducts_eilenberg_moore (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)) (pr12 (ob_of_mnd m)) (pr22 (endo_of_mnd m))) m) (em_ump_1_mor (pr1_of_mnd_total_bicat m) (pr2 (has_em_bicat_of_univ_cats (pr1_of_mnd_total_bicat m))) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) m q))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))
q: em_cone m

preserves_bincoproduct (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))
q: em_cone m
preserves_bincoproduct (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) m q)))
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))
q: em_cone m

preserves_bincoproduct (Monads.functor_from_Monad (MonadsInBicatOfUnivCats.mnd_bicat_of_univ_cats_to_Monad (pr1_of_mnd_total_bicat m)))
exact (pr22 (endo_of_mnd m)).
m: mnd (total_bicat (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)))
q: em_cone m

preserves_bincoproduct (mor_of_mnd_mor (mor_of_em_cone (pr1_of_mnd_total_bicat m) (pr1_of_em_cone (disp_subbicat (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) (λ (C : bicat_of_univ_cats) (_ : (λ C0 : bicat_of_univ_cats, BinCoproducts (pr1 C0)) C), identity_preserves_bincoproduct C) (λ (x y z : bicat_of_univ_cats) (Px : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) x) (Py : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) y) (Pz : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) z) (f : bicat_of_univ_cats ⟦ x, y ⟧) (g : bicat_of_univ_cats ⟦ y, z ⟧) (HF : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) x y Px Py f) (HG : (λ (C₁ C₂ : bicat_of_univ_cats) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₁) (_ : (λ C : bicat_of_univ_cats, BinCoproducts (pr1 C)) C₂) (F : bicat_of_univ_cats ⟦ C₁, C₂ ⟧), preserves_bincoproduct F) y z Py Pz g), composition_preserves_bincoproduct HF HG)) m q)))
exact (pr22 (mor_of_mnd_mor (mor_of_em_cone m q))). Defined.